Covered in lectures. Check back once the chapter is concluded.
10 Residue theorem
Definition 10.1 Let \(z_0\) be an isolated singularity of a holomorphic function \(f\colon U\to\C.\) The residue \(\Res_{z_0}(f)\) is the residue of the Laurent series Equation 9.7. In other words, \[f(z)=\sum_{n=-\iy}^\iy a_n(z-z_0)^n\implies \Res_{z_0}(f)=a_{-1}.\]
Proposition 10.1 Let \(z_0\) be an isolated singularity of both the holomorphic functions \(f,g\colon U\to\C.\)
- \(\Res_{z_0}(f+g)=\Res_{z_0}(f)+\Res_{z_0}(g)\)
- \(\Res_{z_0}(\la f)=\la\Res_{z_0}(f)\) for all \(\la\in\C\)
- If \(z_0\) is a pole of order \(m,\) then \(\Res_{z_0}(f)=\frac{1}{(m-1)!}\lim_{z\to z_0}h^{(m-1)}(z)\) for \(h(z)=(z-z_0)^mf(z).\)
Proof.
Example 10.1
Covered in lectures. Check back once the chapter is concluded.
Example 10.2
Covered in lectures. Check back once the chapter is concluded.
Definition 10.2 Let \(\ga\colon[a,b]\to\C\) be a closed piecewise C1 curve and let \(z_0\in\C\setminus\ga([a,b]).\) The winding number of \(\ga\) around \(z_0\) is\[W_{z_0}(\ga)=\frac{1}{2\pi i}\int_\ga\frac{dz}{z-z_0}.\]
Example 10.3
Covered in lectures. Check back once the chapter is concluded.
Pick a subdivision \(a=t_0<t_1<\cdots<t_n=b\) such that \(\ga|_{[t_{k-1},t_k]}\) is continuously differentiable. Using branches of the logarithm, we can write \(\ga(t)=z_0+r(t)e^{i\th(t)}\) for \(r(t)=|\ga(t)-z_0|\) and a continuous function \(\th\colon[a,b]\to\R\) with \(\th|_{[t_{k-1},t_k]}\) continuously differentiable. With this notation, the following holds.
Proposition 10.2
- We have \(W_{z_0}(\ga)=\frac{\th(b)-\th(a)}{2\pi},\) which is an integer.
- For closed curves \(\ga_0,\) \(\ga_1\) in \(\C\setminus\{z_0\}\) we have \[W_{z_0}(\ga_0)=W_{z_0}(\ga_1)\iff\text{$\ga_0,$ $\ga_1$ are homotopic in $\C\setminus\{z_0\}.$}\]
Proof.
Covered in lectures. Check back once the chapter is concluded.
Theorem 10.1 (Residue theorem) Let \(U\subset\C\) be an open set, \(\ga\colon[a,b]\to U\) a null-homotopic piecewise C1 loop in \(U,\) and \(z_1,\ldots, z_n\in U\setminus\ga([a,b])\) points not on the loop \(\ga.\) If \(f\) is a holomorphic function on \(U\setminus\{z_1,\ldots, z_n\},\) then
\[\frac{1}{2\pi i}\int_\ga f(z)dz=\sum_{k=1}^n W_{z_k}(\ga)\Res_{z_k}(f).\]
Proof.
Covered in lectures. Check back once the chapter is concluded.
10.1 Exercises
Exercise 10.1
Determine the winding numbers \(W_{z_0}(\ga)\) in the following cases.
- \(\ga\colon[0,2\pi]\to\C,\) \(\ga(t)=e^{it},\) for \(z_0=0,2.\)
- \(\ga\colon[2\pi,10\pi]\to\C,\) \(\ga(t)=\begin{cases} te^{it} & \text{if $t\in[2\pi, 6\pi],$}\\ 12\pi - t & \text{if $t\in[6\pi,10\pi],$} \end{cases}\) for \(z_0=0, 20.\)
- \(\ga\colon[0,2\pi]\to\C,\) \(\ga(t)=(2+\cos(t))e^{2it},\) for \(z_0=0,4.\)
Exercise 10.2
Use the residue theorem to compute \[\int_{\partial D_r(0)}\frac{(z+2)^2}{z(z-1)^2}dz\]
for all \(0<r\neq1.\)
Hint: Consider the cases \(r>1\) and \(r<1\) separately.
Exercise 10.3
Apply the residue theorem to the contour \(\partial D_1(0)\) and a suitable holomorphic function \(f(z)\) to compute the integral\[\int_0^{2\pi}\frac{dx}{2+\cos(x)}.\]
Exercise 10.4
Compute the integral \[\int_{0}^{+\iy}\frac{\cos(x)}{\sqrt{x}}dx\] by considering \(f(z)=\frac{e^{iz}}{\sqrt{z}}\) and the following contour \(\ga_{r,R}\) for \(r\to 0,\) \(R\to+\iy.\)
Hint: Recall that \(\int_0^{\iy}e^{-x^2}dx=\sqrt{\pi}/2.\)
Exercise 10.5
Compute the integral \[\int_{-\iy}^{+\iy}\frac{dx}{1+x^6}\]
by applying the residue theorem to \(f(z)=\frac{1}{1+z^6}\) and the following upper semi-circular contour \(\ga_R\) for \(R\to+\iy.\)
